Our second main result for the filtering problem is a conditional compactness result, motivated by the study of statistical solutions of the compressible and incompressible Euler equations in [1]}, [2]}, [3]}. In [2]}, the authors study the forward problem for statistical initial data \({\mu }\)  a probability measure on \(L^2_x := L^2(d)\) . They prove that under Assumption REF , the sequence of discretized approximate solutions \(\mu ^\Delta _t := (S^\Delta _t)_\# {\mu }\)  (push-forward by the discretized solution operator) is compact in \({\mathcal {P}}_1(L^2_x)\) , provided that the following measure of average two-point correlations
\( {S}_2^T(\mu ^\Delta _t;r):=\left(\int _0^T\int _{L^2_x}{S}_2(u;r)^2\, d\mu ^\Delta _t(u)\, dt\right)^{1/2},\) 
